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6.AS.2. TWO (ADJACENT) SQUARES TO A SQUARE
The smaller square often has half the edge of the larger, which connects this with 6.AS.1, but this is not essential. The two squares are usually viewed as one piece, i.e. a P pentomino. These items are dissection proofs of the Theorem of Pythagoras -- see Yates (op. cit. in 6.B, pp. 38-39) for some other examples of this point.

See Ripley's for a similar example, but the 2 x 2 square has a 2, 2, 2 triangle attached to an edge.

Another version has squares of area 1 and 8. The area 8 square is cut into four pieces which combine with the area 1 square to make an area 9 square. I call this the 4 - 5 piece square.
Walther Karl Julius Lietzmann (1880-1959). Der Pythagoreische Lehrsatz. Teubner, (1911, 2nd ed., 1917), 6th ed., 1951. [There was a 7th ed, 1953.] Pp. 23-24 gives the standard dissection proof for the Theorem of Pythagoras. The squares are adjacent and if considered as one piece, the dissection has three pieces. He says it was known to Indian mathematicians at the end of the 9C as the Bride's Chair (Stuhl der Braut). (I always thought this name referred to the figure of the Euclid I, 47 -- ??)

Thabit ibn Qurra (= Thābit ibn Qurra). c875. Gives the standard dissection proof for the Theorem of Pythagoras. The squares are adjacent and if considered as one piece, the dissection has three pieces. [Q. Mushtaq & A. L. Tan; Mathematics: The Islamic Legacy; Noor Publishing House, Farashkhan, Delhi, 1993, pp. 71-72] give this and cite Lietzmann. Greg Frederickson [email of 18 Oct 1996] cites Aydin Sayili; Thabit ibn Qurra's generalization of the Pythagorean theorem; Isis 51 (1960) 35-37.

Abu'l 'Abbas al-Fadhl ibn Hatim al-Narizi (or Annairizi). (d. c922.) Ed. by Maximilian Curtze, from a translation by Gherardo of Cremona, as: Anaritii In decem libros priores Elementorum Euclidis Commentarii, IN: Euclidis Opera Omnia; Supplementum; Teubner, Leipzig, 1899. ??NYS -- information supplied by Greg Frederickson.

Johann Christophorus Sturm. Mathesis Enumerata, 1695, ??NYS. Translated by J. Rogers? as: Mathesis Enumerata: or, the Elements of the Mathematicks; Robert Knaplock, London, 1700, ??NYS -- information provided by Greg Frederickson, email of 14 Jul 1995. Fig. 29 shows it clearly and he attributes it to Frans van Schooten (the Younger, who was the more important one), but this source hasn't been traced yet.

Les Amusemens. 1749. Prob. 216, p. 381 & fig. 97 on plate 8: Réduire les deux quarrés en un seul. Usual dissection of two adjacent squares, attributed to 'Sturmius', a German mathematician, i.e the previous entry.

Ozanam Montucla. 1778. Diverses démonstrations de la quarante-septieme du premier livre d'Euclide, ..., version 2. Fig. 27, plate 4. 1778: 288; 1803: 284; 1814: 241-243; 1840: 123-124. This is a version of the proof that (a + b)² = c² + 4(ab/2), but the diagram includes extra lines which produce the standard dissection of two adjacent triangles.

Crambrook. 1843. P. 4, no. 19: One Square to form two Squares -- ??

E. S. Loomis. The Pythagorean Proposition. 2nd ed., 1940; reprinted by NCTM, 1968. On pp. 194 195, he describes the usual dissection by two cuts as Geometric Proof 165 and gives examples back to 1849, Schlömilch.

Family Friend 2 (1850) 298 & 353. Practical Puzzle -- No. X. = Illustrated Boy's Own Treasury, 1860, Prob. 11, pp. 397 & 437. The larger square has twice the edge of the smaller and is shown divided into four, so this is clearly related to 6.AS.1, though the shape is considered as one piece, i.e. a P-pentomino, to be cut into three parts to make a square.

Magician's Own Book. 1857. To form a square, p. 261. = Book of 500 Puzzles, 1859, p. 75. An abbreviated version of Family Friend. Refers to dotted lines in the figure which are drawn solid.

Charades, Enigmas, and Riddles. 1860: prob. 31, pp. 60 & 65; 1862: prob. 32, pp. 136 & 142; 1865: prob. 576, pp. 108 & 155. Dissect a P-pentomino into three parts which make a square. Usual solution.

Peter Parley, the Younger. Amusements of Science. Peter Parley's Annual for 1866, pp. 139 155.

Pp. 143-144: "To form two squares of unequal size into one square, equal to both the original squares." Usual method, with five pieces. On pp. 146-148, he discusses the Theorem of Pythagoras and shows the dissection gives a proof of it.

P. 144: "To make two smaller squares out of one larger." Cuts the larger square along both diagonals and assembles the pieces into two squares.

Hanky Panky. 1872. To form a square, pp. 116-117. Very similar to Magician's Own Book.

Henry Perigal. Messenger of Mathematics 2 (1873) 104. ??NYS -- described in Loomis, op. cit. above, pp. 104-105 & 214, where some earlier possible occurrences are mentioned. He gives a dissection proof of the theorem of Pythagoras using the shapes that occur in the quadrisection of the square -- Section 6.AR. For sides a < b, perpendicular cuts through the centre are made in the square of side b so they meet the sides at distance (b-a)/2 from a corner. These pieces then fit around the square of side a to make a square of side c.

I invented a hinged version of this, in the 1980s?, which is described in: Greg N. Frederickson; Hinged Dissections: Swinging & Twisting; CUP, 2002, pp. 33-34. I am shown demonstrating this on Frederickson's website: www.cs.purdue.edu/homes/gnf/book2/Booknews2/singm.html .

I have seen the assembly of these four pieces and the square of edge a into the square on the hypotenuse in a photo of the Tomb of Ezekiel in the village of Al-Kifil, near Hillah, Iraq.

Mittenzwey. 1880.

Prob. 176, pp. 34 & 85; 1895?: 201, pp. 38 & 88; 1917: 201, pp. 35 & 84. Use the 10 pieces of 6.AS.1, as in Les Amusemens, to make squares of edge 1 and edge 2.

Prob. 180, pp. 34 & 86; 1895?: 205, pp. 39 & 89; 1917: 205, pp. 35-36 & 85. Cut a 2 x 2 and a 4 x 4 into five pieces which make a square. Both the problem and the solutions are inaccurately drawn. The smaller square has a 1, 2, 5 cut off, as for 6.AS.1. The larger square has the same cut off at the lower left and a 2, 4, 25 cut off at the lower right -- these two touch at the midpoint of the bottom edge -- leaving a quadrilateral with edges 4, 25, 5, 3 and two right angles. This is a variant of the standard five piece method.

Alf. A. Langley. Letter: Three-square puzzle. Knowledge 1 (9 Dec 1881) 116, item 97. Cuts two squares into five pieces which form a single square.

Alexander J. Ellis. Letter: The three-square puzzle. Knowledge 1 (23 Dec 1881) 166, item 146. Usual dissection of two adjacent squares, considered as one piece, into three parts by two cuts, which gives Langley's five pieces if the two squares are divided. Suppose the two squares are on a single piece of paper and are ABCD and DEFG, with E on side CD of the larger square ABCD. He notes that if one folds the paper so that B and F coincide, then the fold line meets the line ADG at the point H such that the desired cuts are BH and HF.

R. A. Proctor. Letter or editorial reply: Three square puzzle. Knowledge 1 (30 Dec 1881) 184, item 152. Says there have been many replies, cites Todhunter's Euclid, p. 266 and notes the pieces can be obtained by flipping the large square over and seeing how it cuts the two smaller ones.

R. A. Proctor. Our mathematical column: Notes on Euclid's first book. Knowledge 5 (2 May 1884) 318. "The following problem, forming a well-known "puzzle" exhibits an interesting proof of the 47th proposition." Gives the usual three piece form, as in Ellis.

B. Brodie. Letter: Superposition. Knowledge 5 (30 May 1884) 399, item 1273. Response to the above, giving the five piece version, as in Langley.

Hoffmann. 1893. Chap. III, no. 11: The two squares, pp. 93 & 125 126 = Hoffmann-Hordern, pp. 82-83, with photo. Smaller square has half the edge. The squares are viewed as a single piece. Photo on p. 83 shows The Five Squares Puzzle in paper with box, by Jaques & Son, 1870 1895, and an ivory version, with box, 1850-1900.

Loyd. Tit Bits 31 (3, 10 & 31 Oct 1896) 3, 25 & 75. General two cut version.

Herr Meyer. Puzzles. The Boy's Own Paper 19 (No. 937) (26 Dec 1896) 206 & (No. 948) (13 Mar 1897) 383. As in Hoffmann.

Benson. 1904. The two square puzzle, pp. 192 193.

Pearson. 1907. Part II, no. 108: Still a square, pp. 108 & 182. Smaller square has half the edge.

Loyd. Cyclopedia. 1914. Pythagoras' classical problem, pp. 101 & 352. c= SLAHP, pp. 15 16 & 88. The adjacent squares are viewed as one piece of wood to be cut. Uses two cuts, three pieces.

Williams. Home Entertainments. 1914. Square puzzle, p. 118. P-pentomino to be cut into three pieces to make a square. No solution given.

A. W. Siddons. Note 1020: Perigal's dissection for the Theorem of Pythagoras. MG 16 (No. 217) (Feb 1932) 44. Here he notes that the two cutting lines of Perigal's 1873 dissection do not have to go through the centre, but this gives dissections with more pieces. He shows examples with six and seven pieces. These cannot be hinged.

The Bile Beans Puzzle Book. 1933.

No. 37: No waste. Consider a square of side 2 extended by an isosceles triangle of hypotenuse 2. Convert to a square using two cuts.

No. 39: Square building. P-pentomino to square in two cuts.

Slocum. Compendium. Shows 4 - 5 piece square from Johnson Smith catalogue, 1935.

M. Adams. Puzzle Book. 1939. Prob. C.120: One table from two, pp. 154 & 185. 3 x 3 and 4 x 4 tiled squares to be made into a 5 x 5 but only cutting along the grid lines. Solves with each table cut into two pieces. (I think there are earlier examples of this -- I have just added this variant.)

Ripley's Puzzles and Games. 1966. Pp. 58-59.

Item 5. Two joined adjacent squares to a square, using two cuts and three pieces.

Item 6. Consider a 2 x 2 square with a 2, 2, 2 triangle attached to an edge. Two cuts and three pieces to make a square.